Imaging of Fractal Profiles

In this paper, a model for radar images of fractal (topologically 1-D) profiles is introduced. A twofold approach is followed: on one hand, we analytically solve the problem whenever small-slope profiles are in order; on the other hand, we present a partly analytical and partly numerical setup to cope with the general-slope case. By means of the analytical approach, we evaluate in closed form both the structure function and the power density spectrum of the radar signal. An appropriately smoothed (physical) fractional Brownian model (fBm) process is employed; its introduction is justified by the finite sensor resolution. A fractal scattering model is employed. It is shown that for a fractal profile modeled as an fBm stochastic process, the backscattered signal turns out to be strictly related to the associated fractional Gaussian noise process if a small-slope regime for the observed profile can be assumed. In the analytical-numerical framework, a profile with prescribed fractal parameters is first synthesized; then, fractal scattering methods (applicable to wider slope regimes with respect to the previous case) are employed to compute the signal backscattered toward the sensor. Finally, the power density spectrum of the acquired radar image is estimated. The obtained spectra are favorably compared with the theoretical results, and a parametric study is performed to assess the overall method behavior.